Rearranging Equations Worksheets - Printable Rearranging Equations ... - Free Printable

Since I can't view or access images directly, I can't see the specific task in your uploaded image. However, based on the text you've provided, it appears to be a worksheet titled "Rearranging Formulae (A)" from CGP Maths Resources, which asks students to rearrange each formula to express $ x $ in terms of $ y $.

Here’s how we can solve such problems step-by-step, using examples from the list:

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General Strategy:

To rearrange a formula to make $ x $ the subject:
1. Isolate the term containing $ x $.
2. Use inverse operations (addition/subtraction, multiplication/division, powers/roots) to solve for $ x $.
3. Simplify as needed.

Let me go through a few sample questions from the list and show how to solve them.

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Example 1:

$ y = x + 5 $

We want $ x $ in terms of $ y $.
Subtract 5 from both sides:
$$
x = y - 5
$$

✔ Answer: $ x = y - 5 $

---

Example 2:

$ y = x - 4 $

Add 4 to both sides:
$$
x = y + 4
$$

✔ Answer: $ x = y + 4 $

---

Example 3:

$ y = 6 - x $

We want $ x $. First, subtract 6 from both sides:
$$
y - 6 = -x
$$
Now multiply both sides by $-1$:
$$
x = 6 - y
$$

✔ Answer: $ x = 6 - y $

---

Example 4:

$ y = 9x $

Divide both sides by 9:
$$
x = \frac{y}{9}
$$

✔ Answer: $ x = \frac{y}{9} $

---

Example 5:

$ y = -3x $

Divide both sides by $-3$:
$$
x = -\frac{y}{3}
$$

✔ Answer: $ x = -\frac{y}{3} $

---

Example 6:

$ y = \frac{x}{2} $

Multiply both sides by 2:
$$
x = 2y
$$

✔ Answer: $ x = 2y $

---

Example 7:

$ y = \frac{x}{4} $

Multiply both sides by 4:
$$
x = 4y
$$

✔ Answer: $ x = 4y $

---

Example 8:

$ y = 2x + 5 $

Subtract 5:
$$
y - 5 = 2x
$$
Divide by 2:
$$
x = \frac{y - 5}{2}
$$

✔ Answer: $ x = \frac{y - 5}{2} $

---

Example 9:

$ y = 8x - 7 $

Add 7:
$$
y + 7 = 8x
$$
Divide by 8:
$$
x = \frac{y + 7}{8}
$$

✔ Answer: $ x = \frac{y + 7}{8} $

---

Example 10:

$ y = 10 - 3x $

Subtract 10:
$$
y - 10 = -3x
$$
Divide by $-3$:
$$
x = \frac{10 - y}{3}
$$
(Alternatively: $ x = -\frac{y - 10}{3} $)

✔ Answer: $ x = \frac{10 - y}{3} $

---

Example 11:

$ y = 7(3x + 2) $

First expand:
$$
y = 21x + 14
$$
Subtract 14:
$$
y - 14 = 21x
$$
Divide by 21:
$$
x = \frac{y - 14}{21}
$$

✔ Answer: $ x = \frac{y - 14}{21} $

---

Example 12:

$ y = -4(x + 11) $

Expand:
$$
y = -4x - 44
$$
Add 44:
$$
y + 44 = -4x
$$
Divide by $-4$:
$$
x = -\frac{y + 44}{4}
$$

✔ Answer: $ x = -\frac{y + 44}{4} $

---

Example 13:

$ y = \frac{x}{4} + \frac{1}{5} $

Subtract $ \frac{1}{5} $:
$$
y - \frac{1}{5} = \frac{x}{4}
$$
Multiply by 4:
$$
x = 4\left(y - \frac{1}{5}\right) = 4y - \frac{4}{5}
$$

✔ Answer: $ x = 4y - \frac{4}{5} $

---

Example 14:

$ y = 10 - \frac{x}{5} $

Subtract 10:
$$
y - 10 = -\frac{x}{5}
$$
Multiply by $-5$:
$$
x = -5(y - 10) = -5y + 50
$$

✔ Answer: $ x = 50 - 5y $

---

Example 15:

$ y = \frac{x + 7}{5} $

Multiply both sides by 5:
$$
5y = x + 7
$$
Subtract 7:
$$
x = 5y - 7
$$

✔ Answer: $ x = 5y - 7 $

---

Example 16:

$ y = \frac{7}{x + 4} $

Multiply both sides by $ x + 4 $:
$$
y(x + 4) = 7
$$
Divide by $ y $:
$$
x + 4 = \frac{7}{y}
$$
Subtract 4:
$$
x = \frac{7}{y} - 4
$$

✔ Answer: $ x = \frac{7}{y} - 4 $

---

Example 17:

$ y = \frac{d}{b - x} $

This one has $ d $ and $ b $, but we are solving for $ x $ in terms of $ y $. Assuming $ d $ and $ b $ are constants.

Multiply both sides by $ b - x $:
$$
y(b - x) = d
$$
Divide by $ y $:
$$
b - x = \frac{d}{y}
$$
Solve for $ x $:
$$
x = b - \frac{d}{y}
$$

✔ Answer: $ x = b - \frac{d}{y} $

---

Example 18:

$ y = x^2 $

Take square root of both sides:
$$
x = \pm \sqrt{y}
$$

✔ Answer: $ x = \pm \sqrt{y} $ (Note: Two solutions unless context restricts to positive)

---

Example 19:

$ y = 8x^2 $

Divide by 8:
$$
x^2 = \frac{y}{8}
$$
Take square root:
$$
x = \pm \sqrt{\frac{y}{8}} = \pm \frac{\sqrt{y}}{2\sqrt{2}} = \pm \frac{\sqrt{2y}}{4}
$$

✔ Answer: $ x = \pm \frac{\sqrt{2y}}{4} $ or $ x = \pm \sqrt{\frac{y}{8}} $

---

Example 20:

$ y = 81x^2 $

Divide by 81:
$$
x^2 = \frac{y}{81}
$$
Take square root:
$$
x = \pm \frac{\sqrt{y}}{9}
$$

✔ Answer: $ x = \pm \frac{\sqrt{y}}{9} $

---

Example 21:

$ y = \frac{x^2}{9} $

Multiply by 9:
$$
9y = x^2
$$
Take square root:
$$
x = \pm \sqrt{9y} = \pm 3\sqrt{y}
$$

✔ Answer: $ x = \pm 3\sqrt{y} $

---

Example 22:

$ y = x^3 + 7 $

Subtract 7:
$$
y - 7 = x^3
$$
Cube root:
$$
x = \sqrt[3]{y - 7}
$$

✔ Answer: $ x = \sqrt[3]{y - 7} $

---

Example 23:

$ y = 67 - x^2 $

Rearrange:
$$
x^2 = 67 - y
$$
Take square root:
$$
x = \pm \sqrt{67 - y}
$$

✔ Answer: $ x = \pm \sqrt{67 - y} $

---

Example 24:

$ y = \frac{5x^2 - 7}{13} $

Multiply by 13:
$$
13y = 5x^2 - 7
$$
Add 7:
$$
13y + 7 = 5x^2
$$
Divide by 5:
$$
x^2 = \frac{13y + 7}{5}
$$
Take square root:
$$
x = \pm \sqrt{\frac{13y + 7}{5}}
$$

✔ Answer: $ x = \pm \sqrt{\frac{13y + 7}{5}} $

---

Example 25:

$ y = \sqrt{11x} $

Square both sides:
$$
y^2 = 11x
$$
Divide by 11:
$$
x = \frac{y^2}{11}
$$

✔ Answer: $ x = \frac{y^2}{11} $

---

Example 26:

$ y = \sqrt{x + 2} $

Square both sides:
$$
y^2 = x + 2
$$
Subtract 2:
$$
x = y^2 - 2
$$

✔ Answer: $ x = y^2 - 2 $

---

Example 27:

$ y = \sqrt{10 - x} $

Square both sides:
$$
y^2 = 10 - x
$$
Rearrange:
$$
x = 10 - y^2
$$

✔ Answer: $ x = 10 - y^2 $

---

Example 28:

$ y = \frac{8x - 3}{2} $

Multiply by 2:
$$
2y = 8x - 3
$$
Add 3:
$$
2y + 3 = 8x
$$
Divide by 8:
$$
x = \frac{2y + 3}{8}
$$

✔ Answer: $ x = \frac{2y + 3}{8} $

---

Example 29:

$ y = 5 + \frac{6}{2x + 3} $

Subtract 5:
$$
y - 5 = \frac{6}{2x + 3}
$$
Take reciprocal:
$$
\frac{1}{y - 5} = \frac{2x + 3}{6}
$$
Multiply both sides by 6:
$$
\frac{6}{y - 5} = 2x + 3
$$
Subtract 3:
$$
\frac{6}{y - 5} - 3 = 2x
$$
Divide by 2:
$$
x = \frac{1}{2} \left( \frac{6}{y - 5} - 3 \right)
$$

✔ Answer: $ x = \frac{1}{2} \left( \frac{6}{y - 5} - 3 \right) $

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Example 30:

$ y = \frac{8 - x}{4} $

Multiply by 4:
$$
4y = 8 - x
$$
Rearrange:
$$
x = 8 - 4y
$$

✔ Answer: $ x = 8 - 4y $

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## Summary

All these problems follow the same logic:
- Isolate the variable $ x $
- Use inverse operations
- Be careful with fractions, roots, and brackets
- For squares/cubes, use square/cube roots
- Always check if multiple solutions exist (especially with even roots)

If you’d like me to solve all 30 problems in full, or focus on a particular one that's giving you trouble, let me know! Just tell me which ones you need help with.